A073570 -id:A073570 - cp's OEIS frontend

A059358 Coefficients in expansion of Sum_{n >= 1} x^n/(1-x^n)^4.

0, 1, 5, 11, 25, 36, 71, 85, 145, 176, 260, 287, 455, 456, 649, 726, 961, 970, 1376, 1331, 1820, 1866, 2315, 2301, 3175, 2961, 3736, 3830, 4729, 4496, 5966, 5457, 6945, 6842, 8114, 7890, 10196, 9140, 11215, 11126, 13420, 12342, 15730, 14191, 17515, 17106, 19601

Offset: 0

N. J. A. Sloane, Jan 27 2001

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000203, A000292, A001157, A001158, A002129, A007437, A013662, A048272, A073570, A101289.

a:= proc(n) option remember;
      add(d*(d+1)*(d+2)/6, d=numtheory[divisors](n))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 12 2023

With[{nn=50},CoefficientList[Series[Sum[x^n/(1-x^n)^4,{n,nn}],{x,0,nn}],x]] (* Harvey P. Dale, May 14 2013 *)

a(n) = if(n==0, 0, sumdiv(n, d, binomial(d+2, 3))); \\ Seiichi Manyama, Apr 19 2021

a(n) = if(n==0, 0, my(f = factor(n)); (sigma(f, 3) + 3*sigma(f, 2) + 2 * sigma(f)) / 6); \\ Amiram Eldar, Dec 29 2024

a(n) = (1/6)*(sigma_3(n) + 3*sigma_2(n) + 2*sigma_1(n)), i.e., this sequence is the inverse Möbius transform of tetrahedral (or pyramidal) numbers: n*(n+1)(n+2)/6 with g.f. 1/(1-x)^4 (cf. A000292). - Vladeta Jovovic, Aug 31 2002
L.g.f.: -log(Product_{k>=1} (1 - x^k)^((k+1)*(k+2)/6)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 21 2018
From Amiram Eldar, Dec 29 2024: (Start)
Dirichlet g.f.: zeta(s) * (zeta(s-3) + 3*zeta(s-2) + 2*zeta(s-1)) / 6.
Sum_{k=1..n} a(k) ~ (zeta(4)/24) * n^4. (End)

A101289 Inverse Moebius transform of 5-simplex numbers A000389.

Original entry on oeis.org

1, 7, 22, 63, 127, 280, 463, 855, 1309, 2135, 3004, 4704, 6189, 9037, 11776, 16359, 20350, 27901, 33650, 44695, 53614, 68790, 80731, 103776, 118882, 148701, 171220, 210469, 237337, 292292, 324633, 393351, 438922, 522298, 576346, 690333, 749399

Offset: 1

Jonathan Vos Post, Mar 31 2006

From Georg Fischer, Aug 06 2025: (Start)
The general pattern is a(n) = Sum_{d|n} (Product_{k=0..m-1} d+k)/m! = Sum_{d|n} binomial(d+m-1, m) = Sum{d|n} Axxxxxx(d), with:
m  Axxxxxx  resulting sequence
------------------------------
2  A000217  A007437
3  A000292  A116963
4  A000332  A073570
5  A000389  A101289 (this sequence)
The other formulas generalize correspondingly.
A116989 uses A000579 and m=6 within a modified formula.
(End)

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000389, A007437, A013664, A073570, A116963, A116969, A332508.

Cf. A000203, A001157, A001158, A001159, A001160.

a(n) = sumdiv(n, d, binomial(d+4, 5)); \\ Seiichi Manyama, Apr 19 2021

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+4, 5)*x^k/(1-x^k))) \\ Seiichi Manyama, Apr 19 2021

a(n) = my(f = factor(n));  (sigma(f, 5) + 10*sigma(f, 4) + 35*sigma(f, 3) + 50*sigma(f, 2) + 24*sigma(f))/120; \\ Amiram Eldar, Dec 30 2024

a(n) = Sum_{d|n} d*(d+1)*(d+2)*(d+3)*(d+4)/120 = Sum_{d|n} C(d+4,5) = Sum{d|n} A000389(d) = Sum_{d|n} (d^5+10*d^4+35*d^3+50*d^2+24*d)/120.
G.f.: Sum_{k>=1} x^k/(1 - x^k)^6 = Sum_{k>=1} binomial(k+4,5) * x^k/(1 - x^k). - Seiichi Manyama, Apr 19 2021
From Amiram Eldar, Dec 30 2024: (Start)
a(n) = (sigma_5(n) + 10*sigma_4(n) + 35*sigma_3(n) + 50*sigma_2(n) + 24*sigma_1(n)) / 120.
Dirichlet g.f.: zeta(s) * (zeta(s-5) + 10*zeta(s-4) + 35*zeta(s-3) + 50*zeta(s-2) + 24*zeta(s-1)) / 120.
Sum_{k=1..n} a(k) ~ (zeta(6)/720) * n^6. (End)

A332508 a(n) = Sum_{d|n} binomial(n+d-2, n-1).

Original entry on oeis.org

1, 3, 7, 25, 71, 280, 925, 3561, 12916, 49346, 184757, 710255, 2704157, 10427747, 40119781, 155288897, 601080391, 2334714319, 9075135301, 35352181116, 137846759282, 538302226628, 2104098963721, 8233718962365, 32247603703576, 126412458920775, 495918551104687

Offset: 1

Ilya Gutkovskiy, Feb 14 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A000005, A000203, A007437, A059358, A073570, A101289, A308814, A332470.

Table[DivisorSum[n, Binomial[n + # - 2, n - 1] &], {n, 1, 27}]
Table[SeriesCoefficient[Sum[x^k/(1 - x^k)^n, {k, 1, n}], {x, 0, n}], {n, 1, 27}]

a(n) = sumdiv(n, d, binomial(n+d-2, n-1)); \\ Michel Marcus, Feb 14 2020

a(n) = [x^n] Sum_{k>=1} x^k / (1 - x^k)^n.

a(n) ~ 4^(n-1) / sqrt(Pi*n). - Vaclav Kotesovec, Aug 04 2022

A343548 a(n) = Sum_{d|n} binomial(d+n-1,n).

Original entry on oeis.org

1, 4, 11, 41, 127, 498, 1717, 6610, 24366, 93391, 352717, 1358826, 5200301, 20097076, 77562773, 300786339, 1166803111, 4539163784, 17672631901, 68933291834, 269129233484, 1052113994124, 4116715363801, 16124221819056, 63205303242628, 247961973949228, 973469736360283

Offset: 1

Seiichi Manyama, Apr 19 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A000005, A000203, A007437, A059358, A073570, A101289, A332508, A343549.

a[n_] := DivisorSum[n, Binomial[# + n - 1, n] &]; Array[a, 30] (* Amiram Eldar, Apr 25 2021 *)

a(n) = sumdiv(n, d, binomial(d+n-1, n));

a(n) = [x^n] Sum_{k>=1} x^k/(1 - x^k)^(n+1).

a(n) = [x^n] Sum_{k>=1} binomial(k+n-1,n) * x^k/(1 - x^k).

A365439 a(n) = Sum_{k=1..n} binomial(floor(n/k)+4,5).

Original entry on oeis.org

1, 7, 23, 64, 135, 282, 493, 864, 1375, 2166, 3168, 4715, 6536, 9132, 12278, 16525, 21371, 27998, 35314, 44995, 55847, 69504, 84455, 103882, 124428, 150005, 177921, 212017, 247978, 292890, 339267, 395874, 455796, 526692, 600788, 691066, 782457, 891048, 1004814

Offset: 1

Seiichi Manyama, Oct 23 2023

Partial sums of A073570.

Cf. A006218, A024916, A064602, A064603, A064604, A364970, A365409.

a(n) = sum(k=1, n, binomial(n\k+4, 5));

from math import isqrt, comb
def A365439(n): return (-(s:=isqrt(n))**2*comb(s+4,4)+sum((q:=n//k)*(5*comb(k+3,4)+comb(q+4,4)) for k in range(1,s+1)))//5 # Chai Wah Wu, Oct 26 2023

a(n) = Sum_{k=1..n} binomial(k+3,4) * floor(n/k).
G.f.: 1/(1-x) * Sum_{k>=1} x^k/(1-x^k)^5 = 1/(1-x) * Sum_{k>=1} binomial(k+3,4) * x^k/(1-x^k).
a(n) = (A064604(n)+6*A064603(n)+11*A064602(n)+6*A024916(n))/24. - Chai Wah Wu, Oct 26 2023

A343545 a(n) = n * Sum_{d|n} binomial(d+3,4)/d.

Original entry on oeis.org

1, 7, 18, 49, 75, 177, 217, 428, 549, 890, 1012, 1824, 1833, 2849, 3360, 4732, 4862, 7506, 7334, 10810, 11382, 14729, 14973, 22188, 20850, 27482, 29052, 37408, 35989, 50490, 46407, 61824, 62106, 75854, 75390, 101673, 91427, 116033, 117624, 146680, 135792, 179886, 163228, 208208

Offset: 1

Seiichi Manyama, Apr 19 2021

nonn

Cf. A000203, A038040, A073570, A309731, A343544, A343546, A343547.

a[n_] := n * DivisorSum[n, Binomial[# + 3, 4]/# &]; Array[a, 50] (* Amiram Eldar, Apr 25 2021 *)

a(n) = n*sumdiv(n, d, binomial(d+3, 4)/d);

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+3, 4)*x^k/(1-x^k)^2))

G.f.: Sum_{k>=1} k * x^k/(1 - x^k)^5 = Sum_{k>=1} binomial(k+3,4) * x^k/(1 - x^k)^2.

A363695 Expansion of Sum_{k>0} (1/(1-x^k)^5 - 1).

Original entry on oeis.org

5, 20, 40, 90, 131, 265, 335, 585, 755, 1147, 1370, 2155, 2385, 3410, 4042, 5430, 5990, 8295, 8860, 11843, 13020, 16335, 17555, 23125, 23882, 29805, 32220, 39440, 40925, 51644, 52365, 64335, 67450, 79820, 82712, 101575, 101275, 120805, 125830, 148089, 149000, 179490

Offset: 1

Seiichi Manyama, Jun 16 2023

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A007503, A116963, A363628, A363696.

Cf. A073570, A363605, A363608.

a[n_] := DivisorSum[n, Binomial[# + 4, 4] &]; Array[a, 40] (* Amiram Eldar, Jul 05 2023 *)

PARI
```
a(n) = sumdiv(n, d, binomial(d+4, 4));
```

G.f.: Sum_{k>0} binomial(k+4,4) * x^k/(1 - x^k).

a(n) = Sum_{d|n} binomial(d+4,4).

A366934 Expansion of Sum_{k>=1} k^5 * x^k/(1 - x^k)^5.

Original entry on oeis.org

1, 37, 258, 1219, 3195, 9597, 17017, 39338, 63189, 118580, 162052, 316974, 373113, 630959, 826320, 1262692, 1424702, 2353896, 2483414, 3912790, 4397862, 6003569, 6451293, 10240908, 10004850, 13819832, 15382332, 20810398, 20547109, 30847530, 28675527, 40458504, 41853306

Offset: 1

Seiichi Manyama, Oct 29 2023

nonn

Cf. A064987, A366135, A366933.
Cf. A073570, A343545.
Cf. A343573.

a(n) = sumdiv(n, d, d^5*binomial(n/d+3, 4));

a(n) = Sum_{d|n} d^5 * binomial(n/d+3,4).

A366986 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{d|n} binomial(d+k-1,k).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 4, 4, 3, 1, 5, 7, 7, 2, 1, 6, 11, 14, 6, 4, 1, 7, 16, 25, 16, 12, 2, 1, 8, 22, 41, 36, 31, 8, 4, 1, 9, 29, 63, 71, 71, 29, 15, 3, 1, 10, 37, 92, 127, 147, 85, 50, 13, 4, 1, 11, 46, 129, 211, 280, 211, 145, 52, 18, 2, 1, 12, 56, 175, 331, 498, 463, 371, 176, 74, 12, 6

Offset: 1

Seiichi Manyama, Oct 31 2023

			Square  array begins:
  1,  1,  1,  1,   1,   1,   1, ...
  2,  3,  4,  5,   6,   7,   8, ...
  2,  4,  7, 11,  16,  22,  29, ...
  3,  7, 14, 25,  41,  63,  92, ...
  2,  6, 16, 36,  71, 127, 211, ...
  4, 12, 31, 71, 147, 280, 498, ...
  2,  8, 29, 85, 211, 463, 925, ...

Columns k=0..5 give A000005, A000203, A007437, A059358, A073570, A101289.
T(n,n-1) gives A332508.
T(n,n) gives A343548.
Cf. A366977.

T(n, k) = sumdiv(n, d, binomial(d+k-1, k));

G.f. of column k: Sum_{j>=1} x^j/(1 - x^j)^(k+1).

cp's OEIS Frontend

A059358 Coefficients in expansion of Sum_{n >= 1} x^n/(1-x^n)^4.

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A101289 Inverse Moebius transform of 5-simplex numbers A000389.

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A332508 a(n) = Sum_{d|n} binomial(n+d-2, n-1).

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A343548 a(n) = Sum_{d|n} binomial(d+n-1,n).

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A365439 a(n) = Sum_{k=1..n} binomial(floor(n/k)+4,5).

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A343545 a(n) = n * Sum_{d|n} binomial(d+3,4)/d.

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A363695 Expansion of Sum_{k>0} (1/(1-x^k)^5 - 1).

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A366934 Expansion of Sum_{k>=1} k^5 * x^k/(1 - x^k)^5.

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